Georgia Institute of Technology College of Sciences

In this talk I first wish to review my work with Balakrishnan and Muller, giving an algorithm for finding integral points on curves under certain (strong) assumptions. The main ingredients are the theory of p-adic height pairings and the theory of p-adic metrized line bundles. I will then explain a new proof of the main result using a p-adic version of Zhang's adelic metrics, and a third proof which only uses the metric at one prime p. At the same time I will attempt to explain why I think this last proof is interesting, being an indication that there may be new p-adic methods for finding integral points.